A particle of mass in a harmonic oscillator potential has energy eigenstates ,

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

The Structure of Quantum Mechanics, Ch Observables, spectra and the spectral theorem

Problem

A particle of mass mm in a harmonic oscillator potential has energy eigenstates ∣n⟩|n\rangle, n=0,1,2,…n=0,1,2,\dots, with H^∣n⟩=En∣n⟩\hat H|n\rangle=E_n|n\rangle and En=(n+12)ℏωE_n=(n+\tfrac12)\hbar\omega, where ℏω=0.30 eV\hbar\omega=0.30\,\text{eV}. At some instant the particle is prepared in the state ∣ψ⟩=15(2∣0⟩+∣1⟩),|\psi\rangle=\frac{1}{\sqrt5}\big(2|0\rangle+|1\rangle\big), a superposition confined entirely to the ground and first excited levels. (a) Write H^\hat H in spectral form on the two-dimensional subspace spanned by ∣0⟩|0\rangle and ∣1⟩|1\rangle, giving E0E_0 and E1E_1 in eV. (b) Use the spectral theorem to construct H^−1=∑nEn−1∣n⟩⟨n∣\hat H^{-1}=\sum_n E_n^{-1}|n\rangle\langle n| on this subspace, and compute ∣χ⟩=H^−1∣ψ⟩|\chi\rangle=\hat H^{-1}|\psi\rangle, giving its coefficients on ∣0⟩|0\rangle and ∣1⟩|1\rangle in eV−1\text{eV}^{-1} to three significant figures. (c) Check the result independently: write ∣χ⟩=a∣0⟩+b∣1⟩|\chi\rangle=a|0\rangle+b|1\rangle with unknowns a,ba,b, impose H^∣χ⟩=∣ψ⟩\hat H|\chi\rangle=|\psi\rangle directly, and solve for a,ba,b.

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